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A sequence (xi) of elements of a
Hilbert space, ℋ, is a basis for ℋ if every h ∈ℋ has a unique, norm-convergent
expansion of the form h = ∑
aixi, where (ai) is a sequence of scalars. The sequence
is minimal if there exists a sequence (yi) ⊂ℋ such that (xi,yj) = δij. Every
basis is minimal, and the sequence (ai) in the expansion of h (above) is
given by ai = (h,yi). In this paper, we restrict our attention to real Hilbert
space.
We derive, from classical characterizations of bases in B-spaces, criterea for (xi)
to be a basis for ℋ, as well as for (xi) to be minimal in ℋ. We show that the
sequence is minimal if and only if there are sequences (gi) ⊂ℋ whose Gram matrices
have a prescribed form. Similar conditions are obtained for (xi) to be a basis for
ℋ.
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